Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 29, 1425 - 1432
A Maple Program for Solving Systems of Linear and Nonlinear Integral Equations by Adomian Decomposition Method Jafar Biazar and Masumeh Pourabd Department of Mathematics Guilan University Rasht, P.O.Box 1914, Iran
[email protected]
Abstract. Mathematical modeling for many problems in different disciplines, such as engineering, chemistry, physics and biology leads to integral equation, or system of integral equations. It’s the reason of great interest for solving these equations. There are some analytical and numerical methods for solving Volterra integral equations, but extension of these methods to systems of such integral equations is not easy to employ. Adomian decomposition method, well address in [1,2] has been used to solved some of these systems such as systems of differential equations, systems of integral equations and even systems of integro-differential equation [3,4,5]. Applying this method needs some computations which is sometimes boring, having a program to do all computations would be interesting and helpful. In this article a maple program is prepared to solve a system of Volterra integral equations of the second kind, linear or non-linear. Keywords: Adomian decomposition, Systems of Volterra integral equations of second kind 1. Introduction Consider the following system of second kind Volterra integral equations: t F (t) = g(t) + G(s, t, F (s))ds, (1.1) 0
Where
F (t) = (f1 (t), f2 (t), . . . , fn (t))t , G(t) = (g1 (t), g2 (t), . . . , gn (t))t ,
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J. Biazar and M. Pourabd
G(s, t, F (s)) = (G1 (s, t, F (s)), G2(s, t, F (s)), . . . , Gn (s, t, F (s))). Especial case of linear systems comes from linearity of the function V (s, t, F (s)). Consider the ith equation in (1), Fi (t) = gi(t) +
t
0
Gi (s, t, f1 (t), f2 (t), . . . , fn (t))ds,
(1.2)
Adomian decomposition method considers the Solution as the summation of a series say:
Fiλ =
∞
fij (t)λj
(1.3)
j=0
and Gi (s, t, f1 (t), f2 (t), . . . , fn (t)), as a series, say: Gi (s, t, f1 (t), f2 (t), . . . , fn (t)) =
∞
Aij (f10 , . . . , f1j , f20 , . . . , f2j , . . . , f1j , fn0 , . . . , fnj )λj
(1.4)
j=0
whereAij (f10 , . . . , f1j , f20 , . . . , f2j , . . . , f1j , fn0 , . . . , fnj ), which depends on fk0 , . . . , fkj , k = 1, 2, . . . called Adomian polynomial and are define as: Aij =
∞ ∞ 1 di [ i Giλ (s, t, f1i λi , . . . , fni λi )]λ=0 i! dλ i=0 i=0
(1.5)
2. Maple program Inputs of the algorithm are the following: n,the number of equations in the system of integral equations, m, the number of terms of the approximations to the solutions, gi (t) , the first term in the right hand side of the canonical form, equation (2). Here is the main program: >restart; >with(student): INPUT 1 >G[1]:=G1 (s, t, f1 , f2 , . . . , fN ); G[2]:=G2 (s, t, f1 , f2 , . . . , fN ); .. . G[k]:=GN (s, t, f1 , f2 , . . . , fN )); 2 >g[1]:=f[1];
A maple program for solving systems
g[2]:=f[2] .. . g[N]:=f[N]); 3 >m:=M: 4 >n:=N: 5 > for i from 1 by 1 while i for i from 1 by 1 while i
